CCGPS
Frameworks
Student Edition
Mathematics
Third Grade Unit Two
Operations and Algebraic Thinking:
The Relationship Between Multiplication
and Division
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
Unit 2
The Relationship Between Multiplication and Division
TABLE OF CONTENTS
Unit Overview………..…………………………………….. ..............................................3
Content Standards and What They Could Look Like ......................................................... 4
Practice Standards ................................................................................................................5
Enduring Understanding ......................................................................................................5
Essential Questions ..............................................................................................................6
Concepts & Skills to Maintain .............................................................................................6
Selected Terms and Symbols ...............................................................................................7
Strategies for Teaching and Learning ................................................................................7
Evidence of Learning .........................................................................................................11
One Hundred Hungry Ants! ............................................................................................. 13
What’s My Product? ......................................................................................................... 15
Base Ten Multiplication .....................................................................................................18
Field Day Blunder ..............................................................................................................22
Stamp Shortage ..................................................................................................................25
Sharing Pumpkin Seeds .....................................................................................................28
What Comes First? Chicken? Seeds? ............................................................................... 33
Shake, Rattle, and Roll Revisited ......................................................................................36
Stuck on Division...............................................................................................................39
Division Patterns ................................................................................................................45
Skittles Cupcake Combo ....................................................................................................50
Animal Investigation ..........................................................................................................53
Our Favorite Candy ...........................................................................................................56
Leap Frog ...........................................................................................................................58
Culminating Task.............................................................................................................63
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 2 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
OVERVIEW
In this unit, students will:
 begin to understand the concepts of multiplication and division
 learn the basic facts of multiplication and their related division facts
Students develop an understanding of the meanings of multiplication and division of
whole numbers through activities and problems involving equal-sized groups, arrays, and area
models; multiplication is finding an unknown product, and division is finding an unknown factor
in these situations. For equal-sized group situations, division can require finding the unknown
number of groups or the unknown group size.
Some common misconceptions that students may have are thinking a symbol (? or ) is
always the place for the answer. This is especially true when the problem is written as 15 ÷ 3 =?
or 15 =
of models is essential in helping students eliminate this understanding.
Another key misconception is that the use of a symbol to represent a number once cannot
be used to represent another number in a different problem/situation. Presenting students with
multiple situations in which they select the symbol and explain what it represents will counter
this misconception.e
Unknown Change Unknown Start Unknown
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 3 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
Unknown Product
Equal Groups
Arrays1,
Area2
36=?
There are 3 bags with 6
plums in each bag. How
many plums are there in all?
3  ? – 18, and 18 3 = ?
If 18 plums are shared
equally into 3 bags, then
how many plums will be in
each bag?
Measurement example. You
need 3 lengths of string,
each6 inches long. How
much string will you need
altogether?
Measurement example. You
have 18 inches of string,
which you will cut into 3
equal pieces. How long will
each piece of string be?
There are 3 rows of apples
with 6 apples in each row.
How many apples are there?
If 18 apples are arranged
into 3equal rows, how many
apples will be in each row?
Area example. What is the
area of a 3 cm by 6 cm
rectangle?
Area example. A rectangle
has area 18 square
centimeters. If one side is 3
cm long, how long is a side
next to it?
A red hat costs $18 and that
is3 times as much as a blue
hat costs. How much does a
blue hat cost?
A blue hat costs $6. A red
hat costs 3 times as much as
the blue hat. How much
does the red hat cost?
Compare
General
Group Size Unknown
(“How many in each group?
Division)
Measurement example. A
rubber band is 6 cm long.
How long will the rubber
band be when it is stretched
to be 3times as long?
a b = ?
Measurement example. A
rubber band is stretched to
be18 cm long and that is 3
times as long as it was at
first. How long was the
rubber band at first?
a ? = p, and pa = ?
Number of Groups
Unknown
(“How many groups?”
Division)
?  6 = 18, and 18  6 = ?
If 18 plums are to be packed
6to a bag, then how many
bags are needed?
Measurement example. You
have 18 inches of string,
which you will cut into
pieces that are6 inches long.
How many pieces of string
will you have?
If 18 apples are arranged
into equal rows of 6 apples,
how many rows will there
be?
Area example. A rectangle
has area 18 square
centimeters. If one side is 6
cm long, how long is a side
next to it?
A red hat costs $18 and a
blue hat costs $6. How many
times as much does the red
hat cost as the blue hat?
Measurement example. A
rubber band was 6 cm long
at first. Now it is stretched
to be18 cm long. How many
times as long is the rubber
band now as it was at first?
?b = p, and pb = ?
Adapted from Common Core State Standards Glossary, pg 89
COMMON MULTIPLICATION AND DIVISION SITUATIONS
1
The language in the array examples shows the easiest form of array problems. A harder form is to use the terms rows and
columns: The apples in the grocery window are in 3 rows and 6 columns. How many apples are in there? Both forms are
valuable.
2Area involves arrays of squares that have been pushed together so that there are no gaps or overlaps, so array problems
include these especially important measurement situations. The first examples in each cell are examples of discrete things.
These are easier for students and should be given before the measurement examples.
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 4 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
STANDARDS FOR MATHEMATICAL CONTENT
Represent and solve problems involving multiplication and division.
MCC.3.OA.1 Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of
objects in 5 groups of 7 objects each.
MCC.3.OA.2 Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the
number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a
number of shares when 56 objects are partitioned into equal shares of 8 objects each.
MCC.3.OA.3 Use multiplication and division within 100 to solve word problems in situations
involving equal groups, arrays, and measurement quantities, e.g., by using drawings and
equations with a symbol for the unknown number to represent the problem.
MCC. 3.OA.4 Determine the unknown whole number in a multiplication or division equation
relating three whole numbers.
Represent and interpret data.
MCC.3.MD.3. Draw a scaled picture graph and a scaled bar graph to represent a data set with
several categories. Solve one- and two-step “how many more” and “how many less” problems
using information presented in scaled bar graphs. For example, draw a bar graph in which each
square in the bar graph might represent 5 pets.
MCC.3.MD.4. Generate measurement data by measuring lengths using rulers marked with
halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is
marked off in appropriate units— whole numbers, halves, or quarters.
STANDARDS FOR MATHEMATICAL PRACTICE
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
***Mathematical Practices 1 and 6 should be evident in EVERY lesson***
ENDURING UNDERSTANDINGS (Taken From Georgia Frameworks)


Multiplication and division are inverses; they undo each other.
Multiplication and division can be modeled with arrays.
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 5 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2




Multiplication is commutative, but division is not.
There are two common situations where division may be used.
o Partition (or fair-sharing) - given the total amount and the number of equal
groups, determine how many/much in each group
o Measurement (or repeated subtraction) - given the total amount and the amount in
a group, determine how many groups of the same size can be created.
As the divisor increases, the quotient decreases; as the divisor decreases, the quotient
increases.
There is a relationship between the divisor, the dividend, the quotient, and any
remainder.
ESSENTIAL QUESTIONS













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


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How are multiplication and addition alike?
How are multiplication and addition different?
How are multiplication and addition related?
How are multiplication and division related?
How are subtraction and division related?
How can I show data using a line plot graph?
How can multiplication and division be used to solve real world problems?
How can the same array represent both multiplication and division?
How can we connect multiplication facts with their array models?
How can we model division?
How can we model multiplication?
How can we practice multiplication facts in a meaningful way that will help us remember
them?
How can we use patterns to solve problems?
How can we write a mathematical sentence to represent a multiplication model we have
made?
How can we write a mathematical sentence to represent division models we have made?
How do I decide what increment scale to use for a bar graph?
How do I decide what symbol to use when constructing a pictograph?
How do the parts of a division problem relate to each other?
How do you create a bar graph or table?
How do you display collected data?
How do you interpret data in a graph?
How is the commutative property of multiplication evident in an array model?
Is there more than one way of multiplying to get the same product?
Is there more than one way to divide a number to get the same quotient?
What are strategies for learning multiplication facts?
What do the parts of a division problem represent?
What happens to the quotient when the dividend increases or decreases?
What is the relationship between the divisor and the quotient?
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 6 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2


What parts are needed to make a complete chart, table, or graph? (title, labels, etc.)
Why would you organize data in different ways?
CONCEPTS/SKILLS TO MAINTAIN
It is expected that students will have prior knowledge/experience related to the concepts and
skills identified below. It may be necessary to pre-assess in order to determine if time needs to be
spent on conceptual activities that help students develop a deeper understanding of these ideas.
 Odd and even numbers
 Skip counting by twos, threes, fives, and tens
 Determining reasonableness using estimation
 Addition and subtraction as inverse operations
 Basic addition facts
 Making tens in a variety of ways
 Basic subtraction facts
 Modeling numbers using base 10 blocks and on grid paper
 Using addition to find the total number of objects in a rectangular array
SELECTED TERMS AND SYMBOLS
The following terms and symbols are often misunderstood. These concepts are not an
inclusive list and should not be taught in isolation. However, due to evidence of frequent
difficulty and misunderstanding associated with these concepts, instructors should pay particular
attention to them and how their students are able to explain and apply them.
The terms below are for teacher reference only and are not to be memorized by the
students. Teachers should present these concepts to students with models and real life examples.
Students should understand the concepts involved and be able to recognize and/or demonstrate
them with words, models, pictures, or numbers.
 array
 dividend
 division
 divisor
 equal groups
 equations
 factor
 groups of
 measurement division (or repeated subtraction)
 multiplicand
 multiplication
 multiplier
 partial products
 partitioned equally
 product
 quotient
 unknown
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 7 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
STRATEGIES FOR TEACHING AND LEARNING
(Adapted from Common Core Resources, NC Dept. of Public Instruction)
Represent and solve problems involving multiplication and division.
In Grade 2, students found the total number of objects using rectangular arrays, such as a
5 x 5, and wrote equations to represent the sum. This strategy is a foundation for multiplication
because students should make a connection between repeated addition and multiplication.
Students need to experience problem-solving involving equal groups (whole unknown or
size of group is unknown) and multiplicative comparison (unknown product, group size
unknown or number of groups unknown) as shown in the table in the unit overview. No attempt
should be made to teach the abstract structure of these problems.
Encourage students to solve these problems in different ways to show the same idea and
be able to explain their thinking verbally and in written expression. Allowing students to present
several different strategies provides the opportunity for them to compare strategies.
Sets of counters, number lines to skip count and relate to multiplication and arrays/area
models will aid students in solving problems involving multiplication and division. Allow
students to model problems using these tools. They should represent the model used as a drawing
or equation to find the solution.
This shows multiplication using grouping with 3 groups of 5 objects and can be written as 3 × 5.
Provide a variety of contexts and tasks so that students will have more opportunity to
develop and use thinking strategies to support and reinforce learning of basic multiplication and
division facts.
Have students create multiplication problem situations in which they interpret the product
of whole numbers as the total number of objects in a group and write as an expression. Also,
have students create division-problem situations in which they interpret the quotient of whole
numbers as the number of shares.
Students can use known multiplication facts to determine the unknown fact in a
multiplication or division problem. Have them write a multiplication or division equation and the
related multiplication or division equation. For example, to determine the unknown whole
27. They should ask themselves questions such as, “How many 3s are in 27 ?” or “3 times what
number is 27?” Have them justify their thinking with models or drawings.
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 8 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
Represent and interpret data.
Representation of a data set is extended from picture graphs and bar graphs with singleunit scales to scaled picture graphs and scaled bar graphs. Intervals for the graphs should relate
to multiplication and division with 100 (product is 100 or less and numbers used in division are
100 or less). In picture graphs, use values for the icons in which students are having difficulty
with multiplication facts. For example,☺ represents 7 people. If there are three ☺, students
should use known facts to determine that the three icons represents 21 people. The intervals on
the vertical scale in bar graphs should not exceed 100.
Students are to draw picture graphs in which a symbol or picture represents more than
one object. Bar graphs are drawn with intervals greater than one. Ask questions that require
students to compare quantities and use mathematical concepts and skills. Use symbols on picture
graphs that student can easily represent half of, or know how many half of the symbol represents.
Students are to measure lengths using rulers marked with halves and fourths of an inch and
record the data on a line plot. The horizontal scale of the line plot is marked off in whole
numbers, halves or fourths. Students can create rulers with appropriate markings and use the
ruler to create the line plots.
Although intervals on a bar graph are not in single units, students count each square as
one. To avoid this error, have students include tick marks between each interval. Students should
begin each scale with 0. They should think of skip- counting when determining the value of a bar
since the scale is not in single units.
Pictographs: Scaled pictographs include symbols that represent multiple units. Below is an example
of a pictograph with symbols that represent multiple units. Graphs should include a title, categories,
category label, key, and data. How many more books did Juan read than Nancy?
Number of Books Read
Nancy
Juan
= 5 books
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 9 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
Single Bar Graphs: Students use both horizontal and vertical bar graphs. Bar graphs include a title, scale,
scale label, categories, category label, and data.
Number of Books Read
Types of Books Read
30
25
20
15
10
5
0
Nonfiction Biography
Fiction
Mystery
Fairytale
Fantasy
Analyze and Interpret data:
 How many more nonfiction books where read than fantasy books?
 Did more people read biography and mystery books or fiction and fantasy books?
 About how many books in all genres were read?
 Using the data from the graphs, what type of book was read more often than a mystery but less
often than a fairytale?
 What interval was used for this scale?
 What can we say about types of books read? What is a typical type of book read?
 If you were to purchase a book for the class library which would be the best genre? Why?
Students in second grade measured length in whole units using both metric and U.S. customary systems.
It is important to review with students how to read and use a standard ruler including details about halves
and quarter marks on the ruler. Students should connect their understanding of fractions to measuring to
one-half and one-quarter inch. Third graders need many opportunities measuring the length of various
objects in their environment.
This standard provides a context for students to work with fractions by measuring objects to a quarter of
an inch.
Example:
Measure objects in your desk to the nearest ½ or ¼ of an inch, display data collected on a line plot. How
many objects measured ¼? ½? etc. …
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 10 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
EVIDENCE OF LEARNING
By the conclusion of this unit, students should be able to demonstrate the following
competencies:
 use mental math to multiply and divide
 use estimation to determine reasonableness of products and quotients computed
 be able to read, interpret, solve, and compose simple word problems dealing with
multiplication and division
 understand how to use inverse operations to verify accuracy of computation
 understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding
the number that makes 32 when multiplied by 8.
 fluently multiply and divide within 100, using strategies such as the patterns and
relationships between multiplication and division
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 11 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
TASKS
Scaffolding Task
Tasks that build
up to the
constructing task.
Constructing Task
Practice Task
Constructing understanding Games/activities
through deep/rich
contextualized problem
solving tasks
Task Name
One Hundred Hungry Ants!
What’s My Product?
Base Ten Multiplication
Field Day Blunder
Stamp Shortage
Sharing Pumpkin Seeds
What Comes First?
Shake, Rattle, and Roll
Revisited
Stuck on Division
Division Patterns
Skittles Cupcake Combos
Animal Investigation
Our Favorite Candy
Leap Frog
Ice Cream Scoops
Task Type
Grouping Strategy
Scaffolding Task
Individual/Partners
Scaffolding Task
Individual/Partners
Practice Task
Partners
Constructing Task
Partners
Constructing Task
Individual/Partners
Constructing Task
Individual/Partners
Constructing Task
Individual/Partners
Practice Task
Individual/Partners
Scaffolding Task
Individual/Partners
Scaffolding Task
Individual/Partners
Constructing Task
Individual/Partners
Scaffolding Task
Individual/Partners
Constructing Task
Individual/Partners
Constructing Task
Individual/Partners
Culminating Task
Individual
Performance Tasks
Summative
assessment for the
unit.
Skills
Multiplication
Multiplication
Multiplication
Multiplication
Multiplication
Division
Division
Multiplication
Division
Division
Division
Data
Data
Data
Multiplication, Division, and Data
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 12 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
SCAFFOLDING TASK: One Hundred Hungry Ants!
STANDARDS FOR MATHEMATICAL CONTENT
MCC.3.OA.1. Interpret products of whole numbers, e.g., interpret 5 × 7 as the
total number of objects in 5 groups of 7 objects each. For example, describe a
context in which a total number of objects can be expressed as 5 × 7.
STANDARDS FOR MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
(Information from Van de Walle and Lovin, Teaching Student-Centered Mathematics: Grades 35, page 167)
When multiplying whole numbers it is a good idea to establish the meaning of the factors. We
would say that 4 x 5 means that we have 4 sets of 5. The first factor will tell how much of the
second factor you have. This along with simple story problems is a good beginning to
developing the concept of multiplication.
ESSENTIAL QUESTIONS
 What are the strategies for learning multiplication?
 How can we practice multiplication facts in a meaningful way that will help us remember
them?
 How is the commutative property of multiplication evident in an array model?
MATERIALS




Colored tiles or two-sided counters
Linking cubes (100 for groups of 4)
Something to help organize groups such as paper plates, cups, bowls, etc.
One Hundred Hungry Ants, by Elinor J. Pinczes or similar story
GROUPING
Individual/Partners
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 13 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task students determine the factors by creating equal groups of counters/colored tiles.
Part I
Begin the lesson by reading One Hundred Hungry Ants, by Elinor J. Pinczes or similar story.
Discuss the ways the ants reorganized themselves into equal groups. You can begin the
discussion by asking the following question: When the ants were first interrupted, how did they
arrange themselves? (you will want to draw the pattern on the board or have linking cubes
available to demonstrate the first grouping) Write the multiplication sentence next to the model.
Ask students to explain the factors. “Which number represents which part of the model?” At
this point the discussion will develop around groups and how many are in the groups. Continue
discussing and modeling the arrangements that the ants are put into each time they are
interrupted. To emphasize the idea of equal groups, you may want to ask the students, “Why did
the ants not organize into groups of 3 or 6?” Allow students time to struggle with this idea.
Provide groups of 4 with 100 linking cubes and let them investigate this idea.
Part II
After students have had discussions about the ways that the ants have organized
themselves, they will have an opportunity to organize ants of their own. Students will be given
20 counters and asked to arrange them in as many different equal groups as they can. Students
should record their reasoning using pictures, words, and numbers.
QUESTIONS FOR FORMATIVE ASSESSMENT
 How many ways were you able to organize 20 ants?
 Can you think of another way to organize 20 ants?
 What does your number sentence look like?
 How can you explain your picture and number sentence in words?
DIFFERENTIATION
Extension
 Allow students to use different numbers of ants. (24, 36, 42). They should explain their
reasoning using pictures, words, and numbers.
Intervention

Allow students to work in small guided groups and reduce the number of ants to 12
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 14 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
SCAFFOLDING TASK: What’s My Product?
STANDARDS OF MATHEMATICAL CONTENT
MCC.3.OA.1. Interpret products of whole numbers, e.g., interpret 5 × 7 as the
total number of objects in 5 groups of 7 objects each. For example, describe a
context in which a total number of objects can be expressed as 5 × 7.
STANDARDS FOR MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
Traditionally multiplication tables are emphasized when students begin learning about
multiplication. Students are sent home with flash cards without a true understanding of what
multiplication is. This way of learning multiplication can be difficult for students to understand.
Naturally, students make groups and groups of groups. The creation of groups is a way to find
the total of something in the most efficient way. The following activity allows students to build
on their natural ability to form groups and learn multiplication without memorizing facts in
isolation, but as number facts that can be related to each other in a multitude of ways (Frans van
Galen and Catherine Twomey Fosnot, 2007, Context for Learning Mathematics).
ESSENTIAL QUESTIONS
 What are the strategies for learning multiplication?
 How can we practice multiplication facts in a meaningful way that will help us remember
them?
 How is the commutative property of multiplication evident in an array model?
MATERIALS



Colored tiles or two-sided counters
Something to help organize groups such as paper plates, cups, bowls, etc.
“What’s My Product” recording sheet
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 15 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
GROUPING
Individual/Partners
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
This task allows students to interpret products of whole numbers by creating equal groups with
manipulatives.
Task Directions
Part I
Discuss with students how to group objects. Show a container of 20 counters. Discuss with
students an easy way to count the total number of counters in the container. Have students
arrange the counters into equal groups. As students discuss how to put the 20 counters into
groups write their thinking on the board. Explain to students that in a multiplication problem one
number represents the number of groups and the other number represents the number of objects
in a group.
Part II
Provide students with a given a set of counters or tiles to separate into equal groups.
The students will continue to rearrange tiles into different groupings that are equal. As each
group is arranged, write a multiplication fact to match the arrangement. Students will record
their thinking in the “What’s My Product?” recording Sheet.
FORMATIVE ASSESSMENT QUESTIONS




How many ways were you able to organize the number of counters you were given?
Can you think of another way to organize your counters?
What does your number sentence look like?
How can you explain your picture and number sentence in words?
DIFFERENTIATION
Extension

Increase the numbers of counters in the students’ baggies.
Intervention
 Provide smaller numbers of counters and allow students to work with a partner.
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 16 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
Name_____________________Date___________________
What’s My Product?
Directions: Arrange counters into equal groups. Complete the table below with
your arrangements.
Groups
# of Tiles/Counters
Multiplication
Fact
Total
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 17 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
PRACTICE TASK: Base Ten Multiplication
(Inspired by Catherine Twomey Fosnot’s, Young Mathematicians at Work, Constructing Multiplication and Division)
STANDARDS FOR MATHEMATICAL CONTENT
MCC.3.OA.1. Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of
objects in 5 groups of 7 objects each. For example, describe a context in which a total number of
objects can be expressed as 5 × 7.
STANDARDS FOR MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
When students begin multiplication they are just getting used to counting. Before
multiplication, 6 equaled a group of six objects. They also know that 4 equals a group of four
objects. However, to think of 4 X 6 they have to think of the group of six as one unit because
they need to make four sixes. The four is now used to count groups not objects. This is a hard
concept to grasp for students just learning about numbers. Students have to reorganize their
thinking (Frans van Galen and Catherine Twomey Fosnot, 2007, Contexts for Learning
Mathematics).
The following task will give students practice in reorganizing numbers and developing strategies
that allow them to make sense of the mathematics.
ESSENTIAL QUESTIONS
 What are the strategies for learning multiplication?
 How can we practice multiplication facts in a meaningful way that will help us remember
them?
 How is the commutative property of multiplication evident in an array model?
MATERIALS
 Base Ten Blocks, up to 51 cubes and 60 longs per pair (base 10 template has been
provided as well)
 Spinner, numbered 1-9
 Packs of 3” X 5” index cards, from 1 to 9 cards per pack, 1 pack per pair
 Overhead Base Ten Blocks (optional)
 Math journal
GROUPING
Partners
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 18 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task students determine the factors of 100 by creating addition and/or multiplication
models by placing equal number of Base Ten Blocks of a kind on index cards according to the
spin of a spinner. Students then record the number sentences that their model represents.
Part I:
Ask two volunteers to hold out their hands, palms up.
Count out 2 units into each hand. Ask children how they can find the number of units in
the four hands. Lead children to count the units by twos. Write this on the chalkboard as the
addition sentence 2 + 2 + 2 + 2 = 8. Elicit that 2 units in each of 4 hands means that there is a
total of 8 units. Point out that because the same number, 2, is added 4 times, another way of
recording this is with multiplication. Write the multiplication sentence 4 X 2 = 8 on the board.
Read it aloud as “Four groups of two equal eight.” Have students to suggest ways to record 2
cubes in each of 4 hands.
Part II:
Students will work with a partner to determine how many different ways to cover each
card from a pack (prearranged according to the students needs) with equal numbers of Base Ten
Blocks.
Distribute the prearranged packs of index cards to the students. Instruct the students to
determine how many cards are in their packs. They will spread out the cards, then spin a spinner.
The number that was spun will determine the number of unit blocks on each card.
 How many cards?
 How many units on each?
The students will determine the product and record the number sentence in their math
journal.
Next students will clear off their cards and put an equal number of longs in their place.
 How many cards?
 How many longs on each?
Students will determine the product of the longs and record their number sentence in the
math journal. Students will be asked to compare the values they found for the units and for the
same number of longs. What did they notice? Repeat the activity several times. (If you spin the
same number as before, just spin again!)
FORMATIVE ASSESSMENT QUESTIONS
 What pattern are you noticing?
 What is the relationship between the units and the longs?
 How did you determine your product?
 Could you have determined your product another way?
DIFFERENTIATION
Extension
 Use larger numbers with students that are ready for the challenge
Intervention
 Use smaller numbers and allow students to work in a small group under teacher direction.
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 19 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 20 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 21 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
CONSTRUCTING TASK: Field Day Blunder
STANDARDS FOR MATHEMATICAL CONTENT
MCC.3.OA.3 Use multiplication and division within 100 to solve word problems in situations
involving equal groups, arrays, and measurement quantities, e.g., by using drawings and
equations with a symbol for the unknown number to represent the problem.
MCC. 3.OA.4 Determine the unknown whole number in a multiplication or division equation
relating three whole numbers.
STANDARDS FOR MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
Multiplication and division are usually taught separately. However, multiplication and division
should be combined in order for students to see how they are related. “Experiences with making
and counting groups, especially in contextual situations, are extremely useful. Products or
quotients are not affected by the size of numbers as long as the numbers are within the grasp of
the students” (Teaching Student-Centered Mathematics, 2006, John A. Van de Walle and
LouAnn H. Lovin).
ESSENTIAL QUESTIONS
 What are the strategies for learning multiplication?
 How can we practice multiplication facts in a meaningful way that will help us remember
them?
MATERIALS


drawing paper, blocks, any other materials that will help students visualize the problem
“Field Day Blunder” student recording sheet
GROUPING
Partners
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 22 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
Students will follow the directions below from the “Field Day Blunder” recording sheet.
Mrs. Nelson’s third grade class was very excited about the upcoming field day events.
Each third grade class was given a helmet and a sack for the upcoming sack race. Once
the sack race was complete, Mrs. Nelson’s class moved on to the next race. As the
students rushed to the next event, they left all of their helmets and sacks in a big pile.
Christopher and Megan were left to match the helmets with the sacks. Some of the sacks
were for 2 people, and some were for 3 people. There were 24 helmets in all. Christopher
and Megan were able to match all of the helmets to their sacks. How many 2- and 3person sacks could there be?
FORMATIVE ASSESSMENT QUESTIONS
 What combinations of blocks have you tried so far?
 How will you know when you find the right combination?
 Do you think there is more than one right solution for this task? Why do you think
so? How can you find out?
DIFFERENTIATION
Extension
 Using 24, or another appropriate number. Ask students to develop a strategy to solve
the problem. Then allow students to share their strategies.
 Replace 24 chairs with 30, 36 or 72 for students who can work with larger numbers.
Intervention
 Replace 24 with a smaller number such as 12, 18 or 20. Model this task or a similar
one in a small group setting.
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 23 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
Name _______________________________ Date ______________________
Field Day Blunder
Mrs. Nelson’s third grade class was very excited about the upcoming field day events.
Each third grade class was given a helmet and a sack for the upcoming sack race. Once the sack
race was complete, Mrs. Nelson’s class moved on to the next race. As the students rushed to the
next event, they left all of their helmets and sacks in a big pile. Christopher and Megan were left to
match the helmets with the sacks. Some of the sacks were for 2 people, and some were for 3
people. There were 24 helmets in all. Christopher and Megan were able to match all of the helmets
to their sacks. How many 2- and 3-person sacks could there be?
1. Draw pictures to show all the ways you can arrange the sacks and helmets.
2. Label and write matching number sentences for each arrangement.
3. Choose your favorite arrangement and explain why you think it would be the best
arrangement so that every student has a helmet and a sack.
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 24 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
CONSTRUCTING TASK: Stamp Shortage
STANDARDS FOR MATHEMATICAL CONTENT
MCC.3.OA.3 Use multiplication and division within 100 to solve word problems in situations
involving equal groups, arrays, and measurement quantities, e.g., by using drawings and
equations with a symbol for the unknown number to represent the problem.
MCC. 3.OA.4 Determine the unknown whole number in a multiplication or division equation
relating three whole numbers.
STANDARDS FOR MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
Often the first strategy students use to solve multiplication problems is repeated addition. This is
because they are viewing the situation additively. Repeated addition should be seen as a starting
place in the journey to understanding multiplication. (Context for Learning Mathematics, Frans
van Galen and Catherine Twomey Fosnot, 2007. In this task, students explore other strategies to
solve multiplication and division strategies.
ESSENTIAL QUESTIONS
 What are the strategies for learning multiplication?
MATERIALS



drawing paper
money
“Stamp Shortage” recording sheet
GROUPING
Individual/Partners
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 25 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
Students will follow the directions on the “Stamp Shortage” recording sheet. Encourage students
to show their work using pictures, charts, or tables.
The local grocery store has run out of 47¢ stamps. The only stamps left are 2¢, 3¢, and
4¢. How many different combinations of stamps can be used to make 47¢?
FORMATIVE ASSESSMENT QUESTIONS
 How could division help you solve this problem?
 How could multiplication help you solve this problem?
 How could estimation help you solve this problem?
 Is this the only solution? Can you solve it another way?
DIFFERENTIATION
Extension
 Allow students to create their own stamps and vary the prices.
Intervention
 Allow students to work with a small group and provide money and stamps (or stamp cut
outs, counters, etc.) as manipulatives.
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 26 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
Name ______________________________Date _______________
Stamp Shortage
The local grocery store has run out of 47¢ stamps. The only stamps left are 2¢,
3¢, and 4¢. How many different combinations of stamps can be used to make 47¢?
Solve the above problem. Show all your work using drawings, charts, and/or tables.
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 27 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
CONSTRUCTING TASK: Sharing Pumpkin Seeds
STANDARDS OF MATNEMATICAL CONTENT
MCC.3.OA.2 Interpret whole-number quotients of whole numbers, e.g.,
interpret 56 ÷ 8 as the number of objects in each share when 56 objects are
partitioned equally into 8 shares, or as a number of shares when 56 objects
are partitioned into equal shares of 8 objects each.
MCC.3.OA.3 Use multiplication and division within 100 to solve word
problems in situations involving equal groups, arrays, and measurement
quantities, e.g., by using drawings and equations with a symbol for the
unknown number to represent the problem.
STANDARDS OF MATHEMATICAL PRACTICE
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
This task provides students with an opportunity to develop and discuss strategies for dividing
a two- or three-digit number by a one-digit number. Possible strategies students may use to solve
this type of problem include, using base 10 blocks, using their knowledge of multiplication and
inverse operations, or using repeated subtraction. Third grade is students’ first exposure to larger
number division and it is important to allow students time to make sense of this operation, so that
students will continue to be successful with division in later grades.
ESSENTIAL QUESTIONS



How can we divide larger numbers?
What is the meaning of a remainder?
Does a remainder mean the same thing in every division problem?
MATERIALS


“Sharing Pumpkin Seeds” recording sheet
Base 10 blocks or other materials for counting available for students who wish to use
them
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 28 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2

How Many Seeds in a Pumpkin? by Margaret McNamara or similar book
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will decide how to share pumpkin seeds fairly with a group of children.
Comments
This task can be paired with the following science standard: S3L1b. Identify features of
green plants that allow them to live and thrive in different regions of Georgia.
There are many children’s books about pumpkins and pumpkin seeds, any one of them
could be used as an introduction to this task. One book that deals directly with the number of
seeds in a pumpkin is How Many Seeds in a Pumpkin? by Margaret McNamara, Illustrated
by G. Brian Karas.
Task Directions
Students will solve the two sharing problems on the “Sharing Pumpkin Seeds” recording
sheet.
Problem 1
Ben and his 3 friends toasted 80 pumpkin seeds from their pumpkin. How many seeds
will each child get if they share the pumpkin seeds fairly?
Clearly explain your thinking using words, numbers, and/or pictures.
Students may approach the problem 80 ÷4 in a variety of ways. Some students may build
on their understanding of multiplication as the inverse of division to solve the problem.
Example 1
I know 4 x 2 = 8, so 4 x 20 = 80. If I add 4 groups of 20, I know there are a total of 80.
Therefore, each child will get 20 pumpkin seeds.
Other students may build on their understanding of division as repeated subtraction.
Example 2
4 x 10 = 40
80-40 = 40
Each child got 10 pumpkin seeds.
4 x 10 = 40
40– 40 = 0
Each child got 10 more pumpkin seeds.
Each child received a total of 10 + 10 pumpkin seeds or 20 pumpkin
seeds.
Some students may choose to use base 10 blocks to represent the division problem.
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 29 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
Example 3
First I took out blocks equal to 80.
Then I started sharing the ten strips among four groups.
Comments
After students have had plenty of time to develop an understanding of division using a
method that makes sense to them, begin to talk with students about an efficient way to
record the various strategies they now use.
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 30 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
FORMATIVE ASSESSMENT QUESTIONS
 What is your plan to solve this problem?
 How do you know your answer is correct?
 How does this help you answer the question in the problem?
DIFFERENTIATION
Extension
Have students to compare strategies used to solve each problem. Encourage them to look
for similarities and differences in their approaches to the problem and to discuss the
efficiency of each. Ask students to present their findings to the class.
Intervention
Before asking students to solve the problems on the “Sharing Pumpkin Seeds” recording
sheet, be sure students have been able to solve similar problems with two-digit dividends.
TECHNOLOGY CONNECTION
http://mason.gmu.edu/~mmankus/whole/base10/asmdb10.htm#div A site for teachers and
parents provides information on using base 10 blocks to solve division problems with an area
model.
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 31 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
Name ____________________________Date _________________________
Sharing Pumpkin Seeds
Ben and his 3 friends toasted 80 pumpkin seeds from their pumpkin. How
many seeds will each child get if they share the pumpkin seeds fairly? Clearly
explain your thinking using words, numbers, and/or pictures.
Sarah and her 5 friends toasted 96 pumpkin seeds from their pumpkin. How
many seeds will each child get if they share the pumpkin seeds fairly? Clearly
explain your thinking using words, numbers, and/or pictures.
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 32 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
CONSTRUCTING TASK: What Comes First? Chicken? Egg?
STANDARDS OF MATHEMATICAL CONTENT
MCC. 3.OA.4 Determine the unknown whole number in a multiplication or
division equation relating three whole numbers.
STANDARDS OF MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them. 2.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
This task provides students with an opportunity to develop and discuss strategies for dividing
a two- or three-digit number by a one-digit number. Possible strategies students may use to solve
this type of problem include, using base 10 blocks, using their knowledge of multiplication and
inverse operations, or using repeated subtraction. Third grade is students’ first exposure to larger
number division and it is important to allow students time to make sense of this operation, so that
students will continue to be successful with division in later grades.
ESSENTIAL QUESTIONS


How can multiplication and division be used to solve real world problems?
How can we use patterns to solve problems?
MATERIALS
 “What Comes First?” recording sheet
 drawing paper
 interlocking cubes or other manipulative if necessary
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
(adapted from Teaching Children Mathematics, Volume 15, Number 3,
October 2008, p. 160).
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 33 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
Part I
Begin this task by reviewing their understanding of estimation from Unit 1.
Discuss the word “approximate” and how it is used in estimating. Students
should understand that rounding is not the only form of estimation.
Part II
Students will follow the directions on the “What Comes First” recording
sheet. This should be solved using pictures, numbers, and words.
If most hens lay about 4 eggs each week, how many eggs does the
average hen lay in one month? How many chickens would be needed to
produce 50 eggs in one month? How many chickens would be needed to
produce 70 eggs in one month? If 30 eggs were produced in one month,
approximately how many chickens were needed to produce them?
FORMATIVE ASSESSMENT QUESTIONS
 How could you use patterns to help you solve this?
 How would a table be useful in solving this problem?
 How might you use multiplication/division to solve this problem?
 Could you write a number sentence to explain your picture/table?
 How can you use estimation to help you solve this problem?
DIFFERENTIATION
Extension
 You could increase the number of eggs up to 100 that are needed in a month
 Students could determine how many eggs a hen will lay in one year. (2 hens, 3 hens,
etc.)
Intervention
 Decrease the number of eggs needed each month to numbers that 4 will divide into
evenly such as (24, 36, 48)
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 34 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
Name ___________________________
Date ______________________
What Comes First
If most hens lay about 4 eggs each week, how many eggs does the average hen lay
in one month? How many chickens would be needed to produce 50 eggs in one
month? How many chickens would be needed to produce 70 eggs in one month? If
30 eggs were produced in one month, approximately how many chickens were
needed to produce them? Show your thinking using words, pictures, and
numbers.
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 35 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
PRACTICE TASK: Shake, Rattle, and Roll Revisited
STANDARDS FOR MATHEMATICAL CONTENT
MCC.3.OA.1. Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of
objects in 5 groups of 7 objects each. For example, describe a context in which a total number of
objects can be expressed as 5 × 7.
MCC.3.OA.2 Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the
number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a
number of shares when 56 objects are partitioned into equal shares of 8 objects each.
STANDARDS FOR MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
Students are taught to write equations as early as Kindergarten. Variables and equations are
powerful tools in representing mathematical ideas. In this task students are able to use all of
their strategies to figure out products, quotients, and factors. (Teaching Student-Centered
Mathematics, John A. Van de Walle and LouAnn H. Lovin, 2006).
ESSENTIAL QUESTIONS
 What are the strategies for learning multiplication?
 What are the strategies for learning division?
 How can we practice multiplication and division facts in a meaningful way that will help
us remember them?
MATERIALS



drawing paper, blocks, any other materials that will help students visualize the problem
“Shake Rattle and Roll” student game board
2 dice
GROUPING
Partners
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 36 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
Each player takes turns and rolls the number cubes and covers the product or any two factors of
the product. For example, if a player rolls a 2 and an 8, the player could cover 16 (product), 2
(factor), or 8 (factor). If the product or factors has been covered, the player loses a turn. The first
player to cover five squares in a row vertically, horizontally or diagonally wins the game. This
same concept can be used to practice division facts follow the same concept however, change the
numbers on the game board, focus on the divisor, dividend, and quotient.
FORMATIVE ASSESSMENT QUESTIONS
 What multiplication/division strategies are you using?
 What patterns are you noticing?
DIFFERENTIATION
Extension
 Create game boards with larger numbers
Intervention
 Create game boards with smaller numbers and use 1 die
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 37 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
Shake, Rattle and Roll Revisited
Directions: Each player takes turns and rolls the number cubes and covers the product or any two
factors of the product. If the product of factors has been covered, the player loses a turn. The
first player to cover five squares in a row vertically, horizontally or diagonally wins the game. To
practice division facts follow the same concept however, change the numbers on the game board,
focus on the divisor, dividend, and quotient.
24
4
9
3
18
2
20
12
4
4
1
20
12
4
3
25
5
8
2
3
6
4
30
36
1
5
18
4
9
1
18
6
5
16
1
9
25
20
4
25
3
2
5
4
8
5
12
2
1
15
12
6
18
5
24
3
24
8
3
5
4
24
2
15
8
6
9
36
3
18
6
24
8
5
16
25
2
30
6
2
3
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 38 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
SCAFFOLDING TASK: Stuck on Division
STANDARDS FOR MATHEMATICAL CONTENT
MCC.3.OA.2 Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the
number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a
number of shares when 56 objects are partitioned into equal shares of 8 objects each.
STANDARDS FOR MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
Students should clearly understand how to write number sentences and how to follow written
directions before working independently.
One possible solution is shown below:
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 39 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
ESSENTIAL QUESTIONS





How can we model division?
How are multiplication and division related?
How are subtraction and division related?
How can we write a mathematical sentence to represent division models we have made?
Is there more than one way to divide a number to get the same quotient?
MATERIALS




12 connecting cubes per student
“Stuck on Division” task sheet
“Stuck on Division” recording sheet
Divide and Ride by Stuart J. Murphy or similar book
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will experiment with a set of 12 connecting cubes to determine the
division patterns when the dividend is 12.
Comments
You may choose to open this task by reading, discussing, and modeling the events in Divide
and Ride by Stuart J. Murphy. Divide and Ride is a story about dividing a group of children to
ride amusement park rides. Another suitable book about division is One Hundred Hungry Ants
by Elinor J. Pinczes. Focus on the different ways division can be described (separating into equal
groups, repeated subtraction, and inverse of multiplication.)
The three ways of looking at division are closely related and may be difficult for students to
verbalize initially as they make connections between concrete models and their corresponding
number sentences. Therefore, students need multiple experiences using a given number of cubes
to model repeated subtraction, form equal groups, and explain how these two activities are alike
and different. They also need to understand the inverse relationship of multiplication and
division. Help students make connections to the language of mathematics and between visual and
symbolic representations.
Task Directions
Students will follow the directions below from the “Stuck on Division” task sheet.
Use 12 connecting cubes to complete this task.
1. Begin with 12 cubes and remove the same number of cubes over and over again until
there are none left. Remember, you must remove the same number each time. Make
a model of your idea with the cubes.
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 40 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
2. Use the first row of the “Stuck on Division” recording sheet to
a. write about what you did
b. draw a diagram of your model
c. write a subtraction number sentence that describes your model
3. Find a way to separate your cubes into equal groups. How can you show the
dividend, divisor, and quotient with your cubes?
4. Use the second row of the “Stuck on Division” recording sheet to
a. write about what you did
b. draw a diagram of your cube groups
c. write a division number sentence
5. Now think of a multiplication fact whose product is twelve. Can you make groups of
cubes that prove that division is the opposite of multiplication?
6. Use the third row of the “Stuck on Division” recording sheet to
a. write about what you did
b. draw a diagram of your cube groups
c. write the fact family for your diagram
7. Compare your answers with your friends. Did everyone have the same answers?
How can you tell whose solutions are correct?
FORMATIVE ASSESSMENT QUESTIONS
 Can you explain more than one way to think about dividing a number?
 How can you write your model in a number sentence so others will understand your
model?
 How can we show your model as both a division number sentence and a subtraction
number sentence?
DIFFERENTIATION
Extension
Have students to complete the chart with 13 blocks. Ask students to include leftovers in
their explanations, diagrams, and number sentences.
Intervention
Direct instruction in small groups can provide support for students who struggle with
these concepts and can enable them to develop the ability to describe their thinking.
TECHNOLOGY CONNECTION


http://mcq.wrdsb.on.ca/Admin/Documents/WORC/PDFs/LESSON%20PrimaryMath.
pdf
http://www.lessonplanspage.com/MathLAMultiplicationDivisionUsingTheDoorbellR
ang23.htm Both websites above provide teacher resources for the book The Doorbell
Rang by Pat Hutchins.
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 41 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2

http://www.stuartjmurphy.com/activities/activity_ideas.php Stuart Murphy website
with activity suggestions for Divide and Ride. (Click on level 3 and then click on the
title of the book.)
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 42 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
Name _______________________Date _________________________
Stuck on Division
Task Sheet
Use 12 connecting cubes to complete this task.
1. Begin with 12 cubes and remove the same number of cubes over and over
again until there are none left. Remember, you must remove the same
number each time. Make a model of your idea with the cubes.
2. Use the first row of the “Stuck on Division” recording sheet to
a. write about what you did
b. draw a diagram of your model
c. write a subtraction number sentence that describes your model
3. Find a way to separate your cubes into equal groups. How can you show
the dividend, divisor, and quotient with your cubes?
4. Use the second row of the “Stuck on Division” recording sheet to
a. write about what you did
b. draw a diagram of your cube groups
c. write a division number sentence
5. Now think of a multiplication fact whose product is twelve. Can you make
groups of cubes that prove that division is the opposite of
multiplication?
6. Use the third row of the “Stuck on Division” recording sheet to
a. write about what you did
b. draw a diagram of your cube groups
c. write the fact family for your diagram
7. Compare your answers with your friends. Did everyone have the same
answers? How can you tell whose solutions are correct?
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 43 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
Name __________________________ Date _________________________
Stuck on Division
Recording Sheet
Division is…
Diagram
Number Sentence
Repeated subtraction
Separating a whole into
equal groups
The opposite of
multiplication
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 44 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
SCAFFOLDING TASK: Division Patterns
STANDARDS FOR MATHEMATICAL CONTENT
MCC.3.OA.2 Interpret whole-number quotients of whole numbers, e.g.,
interpret 56 ÷ 8 as the number of objects in each share when 56 objects
are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned
into equal shares of 8 objects each.
STANDARDS OF MATHEMATICAL PRACTICE
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
Students should begin to master multiplication facts in connection with division facts. When
we are trying to determine the quotient for 36 ÷ 9, we are often think 9 times what number is
going to give me 36. It is not a separate fact but closely tied together (Teacher Student-Centered
Mathematics, John A. Van de Walle and LouAnn H. Lovin, 2006).
ESSENTIAL QUESTIONS




How do the parts of a division problem relate to each other?
What is the relationship between the divisor and the quotient?
What happens to the quotient when the dividend increases or decreases?
What do the parts of a division problem represent?
MATERIALS
“Division Patterns” recording sheet
GROUPING
Individual/Partner Task
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 45 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will analyze patterns in division.
Comments
You may want to demonstrate how to use the times table chart to determine the answers to
basic division problems if students have not yet learned the division facts. Teaching the
algorithm for long division is not required at this point, it will be addressed later in this unit.
You may want to open this task by reading and discussing, the events in The Doorbell Rang
by Pat Hutchins or similar book. The Doorbell Rang is a story about dividing a batch of cookies
by a varying number of children. Focus on how the number of cookies each child gets changes as
the number of children increases.
Task Directions
Students will follow the directions below from the “Division Patterns” recording sheet.
There are three parts to every division problem: the dividend, the divisor,
and the quotient. Look at the division problem below to understand what these
terms mean:
28
÷
4
=
28
4
is the dividend, the total amount before we divide.
is the divisor, the number of groups we will make or the number of
items in each group.
is the quotient, number of items in each group or the number of groups.
1. Complete the chart.
2. What do you notice about the dividend numbers as you go
from the top of the chart to the bottom of the chart?
3. What do you notice about the divisor numbers as you go
from the top of the chart to the bottom of the chart?
4. What do you notice about the quotient numbers as you go
from the top of the chart to the bottom of the chart?
5. Describe the pattern that shows the relationship between the
dividend, divisor, and quotient.
FORMATIVE ASSESSMENT QUESTIONS
 What is the same about all of the division problems?
 What is different about all of the division problems?
 What do you notice about the quotients of the division problems?
 Can you describe a pattern you see in this task?
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 46 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
DIFFERENTIATION
Extension
Have students experiment with keeping a different part of the division problem constant
such as the quotient or dividend and make predictions about the outcomes. Have students
record their results and describe their conclusions.
Intervention
Use base-ten manipulative pieces or grid paper as necessary for students who may need to
model each division problem.
TECHNOLOGY CONNECTION



http://mcq.wrdsb.on.ca/Admin/Documents/WORC/PDFs/LESSON%20PrimaryMath.pdf
http://www.lessonplanspage.com/MathLAMultiplicationDivisionUsingTheDoorbell
Rang23.htm Both websites above provide teacher resources for the book The
Doorbell Rang by Pat Hutchins.
http://www.softschools.com/math/games/division_practice.jsp Division practice; the
student or teacher can determine the parameters for the divisor, dividend, and
number of problems
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 47 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
Name _________________________ Date ___________________________
Division Patterns
There are three parts to every division problem: the dividend, the
divisor, and the quotient. Look at the division problem below to
understand what these terms mean:
28
÷
4
=
28
is the dividend, the total amount before we divide.
4
is the divisor, the number of groups we will make or the
number of items in each group.
is the quotient, the number of items in each group.
1. Complete the following chart:
Dividend
Divisor
Quotient
4
4
1
8
4
12
4
16
4
20
÷
4
24
4
28
4
32
4
36
4
40
4
=
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 48 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
2. What do you notice about the dividend numbers as you go from the top of the
chart to the bottom of the chart?
___________________________________________________________
___________________________________________________________
___________________________________________________________
________________________________________________
3. What do you notice about the divisor numbers as you go from the top of the
chart to the bottom of the chart?
___________________________________________________________
___________________________________________________________
___________________________________________________________
4. What do you notice about the quotient numbers as you go from the top of the
chart to the bottom of the chart?
___________________________________________________________
___________________________________________________________
___________________________________________________________
5. Describe the pattern that shows the relationship between the dividend, divisor,
and
quotient._____________________________________________________
___________________________________________________________
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 49 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
CONSTRUCTING TASK: Skittles Cupcake Combos
STANDARDS FOR MATHEMATICAL CONTENT
MCC.3.OA.3 Use multiplication and division within 100 to solve word problems in situations
involving equal groups, arrays, and measurement quantities, e.g., by using drawings and
equations with a symbol for the unknown number to represent the problem.
STANDARDS OF MATHEMATICAL PRACTICE
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning
Background Knowledge
When students are given trivial word problems, they often just ask themselves what
operation is called for; the context becomes irrelevant as they manipulate numbers, applying
what they know. True context keeps students focused and interested in making sense of the math.
Students begin to notice patterns and ask questions about what is going in the problem. Then
students begin to defend their math to one another. The following activity allows students to
build on their knowledge of grouping materials in order to divide more efficiently. (Frans van
Galen and Catherine Twomey Fosnot, 2007, Context for Learning Mathematics).
This task assesses students’ understanding of division and their ability to organize data.
ESSENTIAL QUESTIONS

How are multiplication and division related?
MATERIALS
 paper
 graph paper
 counters, interlocking cubes
GROUPING
Individual/Partner Task
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 50 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
Students will follow directions from the “Skittles Cupcake Combo” recording sheet.
I love Skittles and cupcakes! I decided to bake some cupcakes. I put a bag of Skittles, 45
in all, into my batter and baked a dozen cupcakes. Each cupcake had at least three Skittles
and no more than five. What are the different possible combinations of Skittles?
FORMATIVE ASSESSMENT QUESTIONS
 What combinations of blocks have you tried so far?
 How will you know when you find the right combination?
 Do you think there is more than one right solution for this task? Why do you think
so? Do you have a way of finding out?
DIFFERENTIATION
Extension
 Using 45, or another appropriate number. Ask students to develop a strategy to solve
the problem. Then allow students to share their strategies.
 Replace 50, 75, 90 for students who can work with larger numbers.
Intervention
 Replace 45 with a smaller number such as 12, 24, or 36. Model this task or a similar
one in a small group setting.
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 51 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
Name ______________________
Date __________________
Skittles Cupcake Combos
I love Skittles and cupcakes! I decided to bake some cupcakes. I put a bag of Skittles, 45 in all, into
my batter and baked a dozen cupcakes. Each cupcake had at least three Skittles and no more than
five. What are the different possible combinations of Skittles?
1. Draw pictures to show all the ways you can arrange the Skittles.
2. Label and write matching number sentences for each arrangement.
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 52 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
SCAFFOLDING TASK: Animal Investigation
STANDARDS FOR MATHEMATICAL CONTENT
MCC.3.MD.3. Draw a scaled picture graph and a scaled bar graph to represent a data set with
several categories. Solve one- and two-step “how many more” and “how many less” problems
using information presented in scaled bar graphs. For example, draw a bar graph in which each
square in the bar graph might represent 5 pets.
STANDARDS FOR MATHEMATICAL PRACTICE
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
Students should develop questions that can be aligned with data and collect, organize, and
display data in different ways. Data collection should be for a purpose such as answering a
question. The analysis of data should have the agenda of adding information about some aspect
of our world. (Teaching Student-Centered Mathematics, John A. Van de Walle and LouAnn H.
Lovin, 2007). In this activity students will create picture graphs and bar graphs for a data set and
interpret what the data means.
ESSENTIAL QUESTIONS




How do I decide what increment scale to use for a bar graph?
How do you interpret data in a graph?
How can I show data using a line plot graph?
How do I decide what symbol to use when constructing a pictograph?
MATERIALS

Chart paper/graphing paper
GROUPING
Individual/Partner Task
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 53 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
Review creating tally charts with the class. List five or more common animals on the board. The
animals listed could be a part of the students study on habitats. Then have students raise their
hands for an animal he/she would like to know more information about. Write a tally mark for
next to each animal chosen. If necessary, review how to count tally marks and mark them
correctly for counting purposes. Students should record the data placed on the board on their
own tally sheet. Explain to students that they will display the data in another way using a picture
graph. As an example create a chart on the board and label it with habitats. Have students come
up to the board one at a time and draw a smiley face next to a habitat they have visited or would
like to. Ask the students what the title of the picture graph should be. Enter their suggestion
above the graph. Discuss what they notice from the picture graph. Have students make
comparisons between the rows as well as telling the number of faces in each row. Now ask,
"How many votes does each face represent?" [One] Model how to create a legend at the bottom
of the chart.
Create a second chart near the first one, but use the legend
= 3. Ask the students what that
might mean. Each smiley face now stands for three votes. Students will now create their own
pictograph using the data they collected about animals they would like to know more information
about.
FORMATIVE ASSESSMENT QUESTIONS






How did we display our data?
How did we make it easier to count the tallies in the tally graph?
Why did that notation make it easier?
Can you name the categories that we collected data about for the second tally chart?
How did we show what we found out?
What questions can you answer from looking at the tally graph?
DIFFERENTIATION
Extension
 Allow students to survey other classrooms and create a graph based on this new data.
Intervention
 Provide a set of data for students and allow them to work in small groups.
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 54 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
Name ___________________________
Date ______________
Animal Investigation
Using the data gathered on the Tally Chart for “Animal Investigation” create a
pictograph. Be sure to include all the elements of a graph. Answer the questions
that follow.
1. Which animal received the most votes?
2. Which animal received the least amount of votes?
3. How many more students want to investigate the animal with the most amount
of votes than your choice?
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 55 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
CONSTRUCTING TASK: Our Favorite Candy
STANDARDS ADDRESSED
MCC.3.MD.3. Draw a scaled picture graph and a scaled bar graph to
represent a data set with several categories. Solve one- and two-step
“how many more” and “how many less” problems using information
presented in scaled bar graphs. For example, draw a bar graph in which
each square in the bar graph might represent 5 pets.
STANDARDS FOR MATHEMATICAL PRACTICE
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
It is important for students to be able to gather their own data about a topic that is important to
them. When students formulate the questions they want to ask, the data they gather become
more and more meaningful. (Teacher Student-Centered Mathematics, John A. Van de Walle and
LouAnn H. Lovin, 2006). How they organize the data and the techniques for analyzing them
have a purpose. In this task students will collect data based their favorite candy.
ESSENTIAL QUESTIONS




How do I decide what increment scale to use for a bar graph?
How do you interpret data in a graph?
How can I show data using a line plot graph?
How do I decide what symbol to use when constructing a pictograph?
MATERIALS

Chart paper/graphing paper
GROUPING
Individual/Partner Task
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 56 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
Part I
Review with students how to collect data using a tally chart. Explain to students how to count
tallys appropriately. Review the elements of a graph. Create a class graph as a model over
something that is familiar to students; favorite cars, favorite game, etc.
Part II
Students follow the directions on the “Our Favorite Candy” recording sheet. Have students
analyze the chart on the student recording sheet and complete the numbered tasks.
1. Organize the data by making a tally chart below to record the data.
2. Create a bar graph using the tally chart. Be sure to include a title, labels for the x and y
axis, a scale, and accurate bars.
3. Write two statements that you can learn from analyzing (looking at) this data.
FORMATIVE ASSESSMENT QUESTIONS




How do I decide what increment scale to use for a bar graph?
How do you interpret data in a graph?
How can I show data using a line plot graph?
How do I decide what symbol to use when constructing a pictograph?
DIFFERENTIATION
Extension

Have students survey a class for the same information. Have students compare the data
from the original data set to the data they collected from another class.
Intervention

Lessen the amount of data in the table in order to be more manageable for struggling
students.
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 57 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
Name__________________________
Date________________________
Our Favorite Candy
Ryan
Mark
Anthony
Sarah
Jenise
Annittra
Janice
Jasmine
Teresa
Lania
Ronnie
Jeremy
Rick
Khalil
Samantha
Megan
Joanie
Kavon
Stephanie
Skittles
M & M’s
Gummy Bears
Starburst
Snickers Candy Bar
Airheads
Skittles
M & M’s
Airheads
M & M’s
Starburst
M & M’s
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Gummy Bears
M & M’s
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Starburst
Skittles
Skittles
1. Organize the data by making a tally chart below to record the data
2. Create a bar graph using the tally chart. Be sure to include a title, labels for the x and y axis, a
scale, and accurate bars. Use your journal or another sheet of paper.
3. Write two things you can learn from analyzing (looking at) this data. Use complete sentences.
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 58 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
CONSTRUCTING TASK: Leap Frog
(adapted from Baltimore County Public Schools)
STANDARDS FOR MATHEMATICAL CONTENT
MCC.3.MD.4. Generate measurement data by measuring lengths using rulers marked with
halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is
marked off in appropriate units— whole numbers, halves, or quarters.
STANDARDS OF MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
“The need to gather data will come from the class naturally in the course of discussion or from
questions arising in other content areas. Science, of course, is full of measurements and thus
abounds in data requiring analysis. Line plots are useful counts of things along a numeric scale.
One advantage of a line plot graph is that every piece of data is on the graph. ” (Teacher
Student-Centered Mathematics, John A. Van de Walle and LouAnn H. Lovin, 2006).
In this task students will use data gathered from frog jumps to create a line plot graph.
ESSENTIAL QUESTIONS


What parts are needed to make a complete chart, table, or graph? (title, labels, etc.)
Why would you organize data in different ways?
MATERIALS







Student recording sheet
3 x 5 index card
Scissors
Rulers
Masking Tape
Internet Access
Directions for origami frog http://www.ljhs.sandi.net/faculty/mteachworth/avidinformation/origami-frog-lab-avid.pdf
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 59 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
This task is designed to deepen students understanding of collecting and displaying data. In this
task students will measure the leaps of origami frogs to the nearest inch and plot the
measurement on a line plot graph.
Part I
Have students discuss graphs and their purpose. On the board have examples of different types
of graphs third graders are responsible for learning. Have students identify each graph and
discuss each graphs purpose. Students can attach labels to graphs for a visual representation.
Explain to students that they will be creating a line plot graph. Go into detail of what a line plot
graph is and why it is used.
Part II
Tell students that they are going to create a line plot graph showing how far Origami Frogs jump.
Have students watch a brief YouTube video on Frog Jumping Contest to pique their interests.
You can find a video of a contest on http://www.frogtown.org/ . View video prior to lesson to
make sure it is appropriate. After viewing the video have students discuss what they saw.
Display data of a frog jumping contest you “attended”. Ask students how the data was gathered.
Students should mention that the data is in inches. Briefly discuss measuring and using a ruler.
Ask students which type of graph would best fit the data you collected from a frog jumping
contest. If students do not automatically choose line plot graph, discuss the graph they chose and
why it would not be appropriate and then discuss line plot graphs again. Model plotting two
pieces of data from your data sheet.
Part III
Explain to students that they will have their own frog jumping contest by creating Origami frogs
out of paper. You can find a video of how to make an origami frog on
http://www.youtube.com/watch?v=luG7_nzBHjI&feature=fvo&ad=21937675894 . The video is
easy, however, you may want to write some directions down for your students. Also model each
fold of the frog for clarification. Give students a few minutes to practice jumping with their
paper frogs. Break students into groups of four to six. Students should take turns measuring the
distance the frogs jump. Each group needs masking tape to mark the starting and end point, a
ruler to measure the jump, and a recording sheet. Students should not measure their own jumps.
Have students follow the directions on the student recording sheet.
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 60 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
FORMATIVE ASSESSMENT QUESTIONS




What parts are needed to make a complete chart, table, or graph? (title, labels, etc.)
Why would you organize data in different ways?
Why are graphs used to display data?
What is an appropriate tool to use in order to measure in inches?
DIFFERNTIATION
Extension
 Have students create a frog out of larger paper to see if it makes a difference in the
distance the frogs jump. Have students predict the outcome.
Intervention
 Plot distances with students who are struggling. Guide them as they measure and plot
data.
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 61 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
Name ______________________________ Date _____________
Leap Frog
A. Each frog in the group will take one leap. Someone in your group will
measure the distance your frog jumps. Be sure to place a piece of masking
tape on the starting and end point of the jump. Use a ruler and measure the
distance the frog jumps to the nearest inch. Record the distance on the
chart below. Use the information collected in the table to create a group
line plot graph.
Frog Owner
Distance Jumped (nearest inch)
B. Using the data in the table above create a line plot graph for your group. Be
sure to include all the elements of a line plot graph.
C. Create a line plot graph using all the data from each group.
D. Looking at your class data and your group data what conclusions can you
draw? Were there any outliers?
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 62 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
UNIT TWO CULMINATING TASK
PERFORMANCE TASK: ICE CREAM SCOOPS
STANDARDS ADDRESSED
MCC.3.OA.1 Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of
objects in 5 groups of 7 objects each.
MCC.3.OA.2 Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the
number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a
number of shares when 56 objects are partitioned into equal shares of 8 objects each.
MCC.3.OA.3 Use multiplication and division within 100 to solve word problems in situations
involving equal groups, arrays, and measurement quantities, e.g., by using drawings and
equations with a symbol for the unknown number to represent the problem.
MCC.3.OA.4 Determine the unknown whole number in a multiplication or division equation
relating three whole numbers.
MCC.3.MD.3. Draw a scaled picture graph and a scaled bar graph to represent a data set with
several categories. Solve one- and two-step “how many more” and “how many less” problems
using information presented in scaled bar graphs. For example, draw a bar graph in which each
square in the bar graph might represent 5 pets.
MCC.3.MD.4. Generate measurement data by measuring lengths using rulers marked with
halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is
marked off in appropriate units— whole numbers, halves, or quarters.
BACKGROUND KNOWLEDGE
As students begin to work on this task, they need to understand the meaning of the
terms single, double, triple and double-double scoops of ice cream. The term “doubledouble” is another way of saying “quadruple” and you may want to ask students to explain
why this is true.
ESSENTIAL QUESTION

How do estimation, multiplication, and division help us solve problems in everyday life?
MATERIALS

“Ice Cream Scoops” recording sheet
GROUPING
Independent Task
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 63 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this culminating task, students will use multiplication and division to show different ways
they can spend $3.00 on different flavors of ice cream. In the process, they will double, triple, or
quadruple the price for a single scoop of ice cream.
Task Directions
Have students follow the directions on the “Ice Cream Scoops” Recording Sheet.
Part I. Picture Graph
Using the flavors in the table able, survey your classroom to see which flavor is the most liked in
your class. Display your data in a picture graph. Be sure to add all elements of a picture graph.
Part II. Multiplication and Division
The Super Delicious Ice Cream Shop has the very best ice cream in town.
They sell their ice cream in double scoops, triple scoops, or double-double (that’s
four) scoops. The top selling ice creams are listed on the sign below. You have
$3.00 to spend. Don’t worry about tax.
Use words, pictures, and numbers to show all your work as you answer the
questions below. Think about using estimation to help you consider your choices.
Be sure to show your estimation work.
Ice Cream Flavors and Prices for a Single Scoop
Varoom Vanilla
Cha-cha Chocolate
Cheery Cherry
Rockin’ Rocky Road
Striped Strawberry
Kid’s Delight
$0.67
$1.33
$1.04
$1.12
$0.89
$0.98
Rockin’ Rocky Road
$1.12
1. With $3.00, which flavor can you buy, triple Varoom Vanilla, or triple Cheery
Cherry?Stripled
Would Strawberry
you have any money $0.89
left?
2. To spend most of your money, should you buy a double, triple, or double-double
scoop ofKid’s
Rockin’
Rocky Road? How much
Delight
$0.98 money would you have left?
3. Which ice cream flavor can you buy if you order a double-double scoop?
4. On a different day, you and 5 of your friends decide to combine your money. You
have $11.76 total. You all want to order the same ice cream in a double scoop.
Which flavors are you able to buy?
5. You have been saving pennies for a whole year! You have saved 425 pennies. If you
and two of your friends share the pennies fairly, how many pennies will each of you
have to buy ice cream?
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 64 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
FORMATIVE ASSESSMENT QUESTIONS
 How are you using estimation to help you solve this task?
 What math facts would help you solve this problem?
 Can you use an inverse operation to be sure your solution is correct?
DIFFERENTIATION
Extension
Have students make up their own flavors and prices, use different amounts of money,
and write their own Ice Cream Scoops stories to share with their classmates.
Remediation
While fluency with multiplication facts is required of third graders, it is not required
that all facts will be acquired in the first marking period of the school year. You may
want to allow students to use cueing devices like a times table chart during this
performance assessment as needed.
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 65 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
Name ________________________________________ Date ____________________________
Ice Cream Scoops
The Super Delicious Ice Cream Shop has the very best ice cream in town. They
sell their ice cream in double scoops, triple scoops, or double-double (that’s four)
scoops. The top selling ice creams are listed on the sign below. You have $3.00
to spend. Don’t worry about tax.
Use words, pictures, and numbers to show all your work as you answer the
questions below. Think about using estimation to help you consider your choices.
Be sure to show your estimation work.
Ice Cream Flavors and Prices for a Single Scoop
Varoom Vanilla
$0.67
Cha-cha Chocolate
$1.33
Cheery Cherry
$1.04
Rockin’ Rocky Road
$1.12
Striped Strawberry
$0.89
Kid’s Delight
$0.98
Part I. Picture Graph
Using the flavors in the table able, survey your classroom to see which flavor is the most liked in
your class. Display your data in a picture graph. Be sure to add all elements of a picture graph.
Part II. Multiplication and Division
1. With $3.00, which flavor can you buy, triple Varoom Vanilla, or triple Cheery
Cherry? Would you have any money left?
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 66 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
2. To spend most of your money, should you
buy a double, triple, or double-double
scoop of Rockin’ Rocky Road? How
much money would you have left?
5. On a different day, you and 5 of your
friends decide to combine your money. You
have $11.76 total. You all want to order
the same ice cream in a double scoop.
Which flavors are you able to buy?
3. Which ice cream flavor can you buy if
you order a double-double scoop?
4. You have been saving pennies for a whole
year! You have saved 425 pennies. If you
and two of your friends share the pennies
fairly, how many pennies will each of you
have to buy ice cream?
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 67 of 68
All Rights Reserved
Georgia Department of Education
Common Core Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2
3rd Grade Unit 2 Performance Assessment Rubric
Standard ↓
Exceeding
Meeting
Not Yet Meeting
– Multiplication work
– Multiplication
shows use of
calculations are
diagrams, words,
correct
and/or other suitable
– Evidence of
representations for
estimation is shown
demonstrating mastery
– Evidence of estimation
is shown with
explanations
– Multiplication
calculations are
– incorrect or omitted
– No evidence of
estimation
– Work shows all
– Division number
– Division number
CCGPS.3.OA.3 Use
multiplication and division
within 100 to solve word
problems in situations involving
equal groups, arrays, and
measurement quantities, e.g., by
using drawings and equations
with a symbol for the unknown
number to represent the problem.
– Explanations are
thorough and detailed
and include reasoning
as well as multiple
representations to
support conclusions
– Explanations are
logical and use
specific math
vocabulary to
describe
multiplication or
division process
– Explanations are
omitted or illogical
– Explanations do not
describe the process
used to derive an
answer to the
question asked
CCGPS.3.MD.3. Draw a scaled
picture graph and a scaled bar
graph to represent a data set with
several categories. Solve oneand two-step “how many more”
and “how many less” problems
using information presented in
scaled bar graphs. For example,
draw a bar graph in which each
square in the bar graph might
represent 5 pets.
– All data relevant to the – Work shown is
– Work is not shown
solutions of both
organized and
– Work shown is
multiplication and
logically presented
disorganized,
division problems are – Work shown
inaccurate, or fails to
accurately recorded in
supports conclusions
communicate
an organized fashion
about which ice
mathematical ideas
cream to buy
CCGPS.3.OA.1 Interpret
products of whole numbers, e.g.,
interpret 5 × 7 as the total
number of objects in 5 groups of
7 objects each
CCGPS.3.OA.2 Interpret wholedivision sentences
sentence
sentence does not
number quotients of whole
correctly
corresponds to the
correspond to
question asked in
question
numbers, e.g., interpret 56 ÷ 8 as – Thorough explanation
of
remainders
is
given
word
problem.
–
No mention is made
the number of objects in each
– Explanation of all the – Response indicates
of remainder
share when 56 objects are
possible solutions is
the presence or lack – Solution to division
partitioned equally into 8 shares,
given with reasons for
of a remainder and
problem is incorrect
or as a number of shares when 56
which solution is the
what this indicates
best
– Solution to division
objects are partitioned into equal
problem is correct
shares of 8 objects each.
MATHEMATICS  GRADE 3 UNIT 2: Operations and Algebraic Thinking: the Relationship Between Multiplication and Division
Georgia Department of Education
Dr. John D. Barge, State School Superintendent
May 2012  Page 68 of 68
All Rights Reserved